##
**Introduction to analyzable functions and constructive proof of the Dulac conjecture.
(Introduction aux fonctions analysables et preuve constructive de la conjecture de Dulac.)**
*(French)*
Zbl 1241.34003

Actualités Mathématiques. Paris: Hermann, Éditeurs des Sciences et des Arts. (ISBN 2-7056-6199-9). ii, 340 p. (1992).

As is mentioned by the author, the book which in his initial conception was to be limited to a proof of “Dulac’s conjecture” (finiteness of limit cycles for any polynomial vector field on \({\mathbb{R}}^2\)) has changed during its redaction. In its present form, it may be seen as an illustration of several basic techniques of resummation. Above all, it introduces two new classes of functions: the analyzable functions and the cohesive functions, which the author thinks will be useful for many applications. The proof of Dulac’s conjecture is deliberately treated as a “resummation exercise” and occupies just two chapters out of ten (but these two chapters represent half of the book).

Analyzable (real) functions are, roughly speaking, the natural closure of the algebra of germs of analytic functions at (\({\mathbb{R}},\infty \)), relative to the operations \(+,\times,\partial,\circ\) (composition) and their inverses. An analyzable function \(\varphi \) is completely formalisable, i.e. reducible to a formal transeries \({\widetilde \varphi }\). A transeries is a well-ordered sum of transmonomials, which are themselves irreducible piles of real coefficients and symbols: \(+,\times,\circ\), exp, log. In general the transeries \({\widetilde \varphi }\) is divergent, but one can reconstruct the geometric object \(\varphi (z)\) from the formal object \({\widetilde \varphi }(z)\) by a procedure of accelero-resummation. This is a delicate procedure using passage through a finite number of intermediary “models” \({\widetilde \varphi }_i(\xi_i)\), which are mutually related by the acceleration operators, corresponding via the Borel-Laplace transformation to a change in the variable \(z\) from \(z_i\) to \(z_{i+1}\).

Depending on the nature of the acceleration, the function \({\widehat \varphi }_{i+1} (\xi_i)\) associated to \(\varphi_i (z_i)\) is a germ of function, analytic or cohesive, with a unique (but ramified) extension over \({\mathbb{R}}^+\). The class of cohesive functions contains the most regular of Carleman’s quasi-analytic functions, but has regularity properties which were lacking in these latter functions.

These two notions, analyzibility and cohesivity, constitute the fundamental core of the book.

Let us come back to Dulac’s conjecture. This conjecture claims that a vector field \(X\) on \({\mathbb{R}}^2\), with polynomial coefficients, has at most a finite number of limit cycles, i.e. of isolated closed orbits. It suffices to prove that the limit cycles of a vector field (with isolated singularities) cannot accumulate on some polycycle \({\mathcal T}\) (which may be reduced to a point). One has to study the return map \(F\) of this polycycle \({\mathcal T}\) and prove the finiteness of its isolated fixed points. This is proved more generally in the book for any real analytic vector field defined near some polycycle \({\mathcal T}\).

The method is as follows: one writes the return map \(F=G_r\circ \cdots \circ G_1\) as a composition of transition maps near each summit of the polycycle. Next one looks at the formal counterpart \({\widetilde F}={\widetilde G}_r\circ \cdots \circ {\widetilde G}_1\). Depending on the case, the factors \({\widetilde G}_i\) can be written as formal series or rather elementary transeries. On the other hand, the composition \({\widetilde F}\) is a general transeries, made by exponential-logarithm piles of maximum complexity. Nevertheless this transeries is always accelero-summable, so its sum \(F\) is an analyzable function which has just isolated fixed points, when it is different from identity.

The proof is given in two chapters. In the first, a local study, the structure of the factors \({\widetilde G}_i\), \(G_i\) at each summit is meticulously described. The second, a global study, integrates the entire information to arrive at an exhaustive description of \(F\) and of the passage from \({\widetilde F}\) to \(F\). This allows the author to present a defense and illustration of his resummation theory, utilizing many tools and concepts he has introduced: resurgence, alien derivative, accelerations, medianisation, compensators, emanation, transmonomials and transeries, analyzibility, cohesivity and so on, which are applied here, but which have a more general range as well.

The second part of the book contains complements which enlight and extend the methods used to solve Dulac’s problem. For instance, the author establishes the identity between cohesive functions and the weak accelerated functions, and he explains why, in his opinion, the analyzable functions constitute the ultimate limit to the possibility of formalization of function germs. This leads him to discover the fractal character of the natural scale of growth of function germs at infinity, which leads to the mysterious notions of transfinite iterations and the big Cantor, obtained by elimination of growth zones which bring together indiscernable germs.

Analyzable (real) functions are, roughly speaking, the natural closure of the algebra of germs of analytic functions at (\({\mathbb{R}},\infty \)), relative to the operations \(+,\times,\partial,\circ\) (composition) and their inverses. An analyzable function \(\varphi \) is completely formalisable, i.e. reducible to a formal transeries \({\widetilde \varphi }\). A transeries is a well-ordered sum of transmonomials, which are themselves irreducible piles of real coefficients and symbols: \(+,\times,\circ\), exp, log. In general the transeries \({\widetilde \varphi }\) is divergent, but one can reconstruct the geometric object \(\varphi (z)\) from the formal object \({\widetilde \varphi }(z)\) by a procedure of accelero-resummation. This is a delicate procedure using passage through a finite number of intermediary “models” \({\widetilde \varphi }_i(\xi_i)\), which are mutually related by the acceleration operators, corresponding via the Borel-Laplace transformation to a change in the variable \(z\) from \(z_i\) to \(z_{i+1}\).

Depending on the nature of the acceleration, the function \({\widehat \varphi }_{i+1} (\xi_i)\) associated to \(\varphi_i (z_i)\) is a germ of function, analytic or cohesive, with a unique (but ramified) extension over \({\mathbb{R}}^+\). The class of cohesive functions contains the most regular of Carleman’s quasi-analytic functions, but has regularity properties which were lacking in these latter functions.

These two notions, analyzibility and cohesivity, constitute the fundamental core of the book.

Let us come back to Dulac’s conjecture. This conjecture claims that a vector field \(X\) on \({\mathbb{R}}^2\), with polynomial coefficients, has at most a finite number of limit cycles, i.e. of isolated closed orbits. It suffices to prove that the limit cycles of a vector field (with isolated singularities) cannot accumulate on some polycycle \({\mathcal T}\) (which may be reduced to a point). One has to study the return map \(F\) of this polycycle \({\mathcal T}\) and prove the finiteness of its isolated fixed points. This is proved more generally in the book for any real analytic vector field defined near some polycycle \({\mathcal T}\).

The method is as follows: one writes the return map \(F=G_r\circ \cdots \circ G_1\) as a composition of transition maps near each summit of the polycycle. Next one looks at the formal counterpart \({\widetilde F}={\widetilde G}_r\circ \cdots \circ {\widetilde G}_1\). Depending on the case, the factors \({\widetilde G}_i\) can be written as formal series or rather elementary transeries. On the other hand, the composition \({\widetilde F}\) is a general transeries, made by exponential-logarithm piles of maximum complexity. Nevertheless this transeries is always accelero-summable, so its sum \(F\) is an analyzable function which has just isolated fixed points, when it is different from identity.

The proof is given in two chapters. In the first, a local study, the structure of the factors \({\widetilde G}_i\), \(G_i\) at each summit is meticulously described. The second, a global study, integrates the entire information to arrive at an exhaustive description of \(F\) and of the passage from \({\widetilde F}\) to \(F\). This allows the author to present a defense and illustration of his resummation theory, utilizing many tools and concepts he has introduced: resurgence, alien derivative, accelerations, medianisation, compensators, emanation, transmonomials and transeries, analyzibility, cohesivity and so on, which are applied here, but which have a more general range as well.

The second part of the book contains complements which enlight and extend the methods used to solve Dulac’s problem. For instance, the author establishes the identity between cohesive functions and the weak accelerated functions, and he explains why, in his opinion, the analyzable functions constitute the ultimate limit to the possibility of formalization of function germs. This leads him to discover the fractal character of the natural scale of growth of function germs at infinity, which leads to the mysterious notions of transfinite iterations and the big Cantor, obtained by elimination of growth zones which bring together indiscernable germs.

Reviewer: Robert Roussarie (MR1399559)

### MSC:

34-02 | Research exposition (monographs, survey articles) pertaining to ordinary differential equations |

34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |

40A30 | Convergence and divergence of series and sequences of functions |

44A10 | Laplace transform |